Problem: Simplify; express your answer in exponential form. Assume $a\neq 0, y\neq 0$. $\dfrac{{(a^{3}y^{-4})^{-2}}}{{(a^{3}y^{-4})^{-3}}}$
Explanation: To start, try simplifying the numerator and the denominator independently. In the numerator, we can use the distributive property of exponents. ${(a^{3}y^{-4})^{-2} = (a^{3})^{-2}(y^{-4})^{-2}}$ On the left, we have ${a^{3}}$ to the exponent ${-2}$ . Now ${3 \times -2 = -6}$ , so ${(a^{3})^{-2} = a^{-6}}$ Apply the ideas above to simplify the equation. $\dfrac{{(a^{3}y^{-4})^{-2}}}{{(a^{3}y^{-4})^{-3}}} = \dfrac{{a^{-6}y^{8}}}{{a^{-9}y^{12}}}$ Break up the equation by variable and simplify. $\dfrac{{a^{-6}y^{8}}}{{a^{-9}y^{12}}} = \dfrac{{a^{-6}}}{{a^{-9}}} \cdot \dfrac{{y^{8}}}{{y^{12}}} = a^{{-6} - {(-9)}} \cdot y^{{8} - {12}} = a^{3}y^{-4}$